Homogenization and exact controllability for problems with imperfect interface

Abstract

In this paper we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually we apply the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.

A Finite Difference Method for Two Dimensional Elliptic Interface Problems with Imperfect Contact

“…Javili [14] developed a thermodynamically consistent theory for general imperfect interfaces and to establish a unified computational framework to model all classes of such interfaces using the finite element method. Monsurro [10,27] studied the asymptotic behavior of a problem modelling the stationary diffusion in a two-component heat conductor, with a contact resistance. The flow of heat through the interface separating the two media is supposed to be proportional to the jump of the temperature field.…”